The area of the region outside r=5sinθ and inside r=15sinθ is 50π square units.
The area of the region outside r=5sinθ and inside r=15sinθ, we need to evaluate the integral of the area element dA over the region of interest. The area element in polar coordinates is given by dA = r dr dθ.
The region of interest is the annular region between the two circles, which is defined by the inequalities:
5sinθ ≤ r ≤ 15sinθ
0 ≤ θ ≤ π
Thus, the area of the region is given by:
A = [tex]\int\int dA = \int_0^\pi \int_5sin\theta^{(15sin\theta)} r dr d\theta[/tex]
Using the limits of integration, we can rewrite the integral as:
A = [tex]\int_0^\pi [1/2 (15sin\theta)^2 - 1/2 (5sin\theta)^2] d\theta[/tex]
Simplifying the integrand, we get:
A = [tex]1/2 \int_0^\pi (225sin^2\theta - 25sin^2\theta) d\theta[/tex]
A = [tex]1/2 \int_0^\pi 200sin^2\theta d\theta[/tex]
Using the identity sin²θ = 1/2 - 1/2cos2θ, we get:
A =[tex]1/2 \int_0^\pi 100 - 100cos2\theta d\theta[/tex]
Integrating, we get:
A = 1/2 [100θ - 50sin2θ] from 0 to π
A = 1/2 [100π - 0] - 1/2 [0 - 0]
A = 50π
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The area of the region outside the polar curve r = 5sinθ and inside the polar curve r = 15sinθ is 50π square units.
To calculate the area between two polar curves, we integrate the outer curve and subtract the integral of the inner curve over the desired interval. In this case, the curves are r = 5sinθ and r = 15sinθ, and we want to find the area from θ = 0 to θ = π.
The equation r = 5sinθ represents the inner curve, and r = 15sinθ represents the outer curve.
Using the formula for the area between two polar curves, the area A can be calculated as follows:
A = (1/2) ∫[θ1,θ2] (r_outer^2 - r_inner^2) dθ
Substituting the given equations, we have:
A = (1/2) ∫[0,π] ((15sinθ)^2 - (5sinθ)^2) dθ
Simplifying the equation further:
A = (1/2) ∫[0,π] (225sin^2θ - 25sin^2θ) dθ
A = (1/2) ∫[0,π] 200sin^2θ dθ
Integrating this equation over the given interval, we get:
A = (1/2) * 200 * ∫[0,π] sin^2θ dθ
Using the identity ∫ sin^2θ dθ = (1/2) * (θ - sinθcosθ), we have:
A = (1/2) * 200 * [(π - sinπcosπ) - (0 - sin0cos0)]
A = (1/2) * 200 * [(π - 0) - (0 - 0)]
A = (1/2) * 200 * π
A = 100π
Finally, we subtract the area enclosed by the inner curve r = 5sinθ to get the area between the curves:
A = 100π - (1/2) * 5^2 * π
A = 100π - 25π
A = 75π
Therefore, the area of the region outside r = 5sinθ but inside r = 15sinθ is 50π square units.
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100 POINTS IF YOU HELP MEE !! AND ILL GIVE BRAINLISTT !! Complete each statement with the number used to create an equivalent fraction. Multiply to by in both the numerator and denominator to get Multiply { by in both the numerator and denominator to get 16 Multiply by in both the numerator and denominator to get 24 Multiply 1 by in both the numerator and denominator to get 16
Answer:
Step-by-step explanation:
8/8 5/5
Find the missing length?
A. 18in
B. 6in
C. 2in
D. 36in
Answer:
B. 6in
Step-by-step explanation:
We can use the Pythagorean theorem [ a² + b² = c² ] to find the missing side.
8² + b² = 10²
64 + b² = 100
b² = 36
b = 6
Best of Luck!
Expanded form of 5*10^-9
Answer:
5,000 + 300 + 20 + 5 = 5,325
Step-by-step explanation:
What is a perfect square? Why is [tex]x^{2} + 8x +16[/tex] a perfect square?
Answer:
It is a perfect square. Explanation below.
Explanation:
Perfect squares are of the form
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
. In polynomials of x, the a-term is always x.(
(
x
+
c
)
2
=
x
2
+
2
c
x
+
c
2
)
x
2
+
8
x
+
16
is the given trinomial. Notice that the first term and the constant are both perfect squares:
x
2
is the square of x and 16 is the square of 4.
So we find that the first and last terms correspond to our expansion. Now we must check if the middle term,
8
x
is of the form
2
c
x
.
The middle term is twice the constant times x, so it is
2
×
4
×
x
=
8
x
.
Okay, we found out that the trinomial is of the form
(
x
+
c
)
2
, where
x
=
x
and
c
=
4
.
Let us rewrite it as
x
2
+
8
x
+
16
=
(
x
+
4
)
2
. Now we can say it is a perfect square, as it is the square of
(
x
+
4
)
.
PLEASE HELP ASAP!
Triangle CAT has vertices C (2,3), A(-2,6) and T(-1,-4). A translation maps the point A to A' (-1,4). What are the coordinates of C' under this translation?
A. (1, -3)
B. (3, 1)
C. (1, 8)
D. (-8, 1)
you add 1 and subtracted 2
so 2 + 1 = 3
3 - 2 = 1
answer: b. (3, 1)
P = 100 - 20 cos (8πt/3) That's (8 pi t /3) where t is the time (in seconds). What is the period of the model? Discuss what the period represents in the context of this model.
Answer:
The period of the model is [tex]\frac{3\pi}{4}[/tex] seconds. The period represents the time needed for the function to complete one cycle.
Step-by-step explanation:
Cosine is a trigonometric function and trigonometric functions are characterized by having a periodical behavior. The period is the time needed for the function to cover an angle of [tex]2\pi[/tex] radians. By this approach we find that:
[tex]\frac{8\pi\cdot t}{3} = \frac{2\pi\cdot t}{T}[/tex] (1)
Where:
[tex]t[/tex] - Time, measured in seconds.
[tex]T[/tex] - Period, measured in seconds.
Then, we solve (1) for [tex]T[/tex]:
[tex]\frac{8}{3} = \frac{2\pi}{T}[/tex]
[tex]T = \frac{6\pi}{8}\,s[/tex]
[tex]T = \frac{3\pi}{4}\,s[/tex]
The period of the model is [tex]\frac{3\pi}{4}[/tex] seconds. The period represents the time needed for the function to complete one cycle.
what is the formula of volume
Answer:
length × width × height.
Step-by-step explanation:
Answer:
length × width × height
Step-by-step explanation:
That is the standard formula for a rectangular shape
What is m = -13/6.5 simplified?
Calculate the value of this expression by writing the decimal factors as fractions, then writing their product as a decimal. Show your reasoning. (0.3)⋅(0.2)
Answer:
(3/10)(2/10) = 6/100 = 0.06
Determine the slope and the y-intercept: y = 3x - 1
Answer:
Slope is 3
y - intercept is -1
Answer:
slope = 3, y- intercept is - 1
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = 3x - 1 ← is in slope- intercept form
with slope m = 3 and y- intercept c = - 1
Which graph represents a proportional relationship? (quickly please :)
Answer:
the first one
Step-by-step explanation:
a proportional graph has to go through the origin (0,0) and the first one is the only one that does
Find the area of the regular octagon.
Answer:
3*4
=12 sq. cm
= 12 cm²
Answer:
48cm^{2}
Step-by-step explanation:
[tex]A=4ah=4*4*3=48 cm^{2}[/tex]
How would I solve this?
Answer:
2.74747741945462
Step-by-step explanation:
i know its long but i think it is correct!:)
Christina earned $510 working part-time at the movie theater. Of her paycheck, 1/ 4 pays for car insurance, 1/ 3 for gas, and 20% for miscellaneous expenses. The remaining amount of her paycheck is divided equally for entertainment and savings. How much does Christina spend each month on entertainment?
Answer:
$55.15
Step-by-step explanation:
1/4 of 510 = 127.5
1/3 of 510 = 170
20% of 510 = 102
510-(127.7+102+170)=510-399.7=110.3
110.3/2=55.15
hope this helps
I NEED THIS RN AS FAST AS POSSIBLE! Your teacher decides to split the class into four equal groups with no more than five students in each group. that represents the number of students in the class.
Answer:
option 1 i think
Step-by-step explanation:
imma just take these 5 points back
Carl is deriving the quadratic equation by completing the square. His work at one of the steps is shown. x + b/na = + or - b^2-4ac. Carl is not sure what the value of n is. What value should he use to continue deriving the quadratic equation correctly? A. 1 B. 2 C. 3 D. 4
Answer:
the correct option is B. 2
Step-by-step explanation:
As given,
x + [tex]\frac{b}{na}[/tex] = ± √b²-4ac
⇒x = -[tex]\frac{b}{na}[/tex] ± √b²-4ac
As e know that the quadratic formula is
x = [tex]\frac{-b + or - \sqrt{b^{2} - 4ac } }{2a}[/tex]
So , by comparing with the formula
we get
n = 2
So, the correct option is B. 2
PLS PLS PLS PLS HELP ME
Exponents work the same way on bases that are _[blank A]_ as they do on bases that are numbers.
When a variable has an exponent, the exponent tells you how many times the variable is _[blank B]_ by itself.
In order to make a negative exponent positive, you must take the _[blank C]_ of the base.
A variable raised to the power of _[blank D]_ is always equal to 1.
Match each blank with the correct item that fills in that blank.
multiplied
0
exponents
reciprocal
1
added
variables
opposite
A - variables
B - multiplied
C - reciprocal?
D - 0
//
I'm not too sure about C. it can either be the reciprocal or the opposite.
If the points at (-4, 3) and (2, 3) are reflected over the x-axis to create two new points, what shape can be formed by the four points? Select the most precise description of the shape.
square
rectangle
triangle
trapezoid
Answer:
square
Step-by-step explanation:
what is the value of a + b(4c + 2a)?
Answer:
Step-by-step explanation:
You can't find an exact value, but you can find an simpler expression by using the distributive property.
a + 4bc + 2ab
And that is as much as it can be reduced.
Jim rides the bus to and from school each day. A one-way trip is 8.12 kilometers. How many kilometers does he travel in 3 days?
Answer:
48.72 km in 3 days.
Step-by-step explanation:
Given that,
A one-way trip is 8.12 kilometers.
We need to find the distance he travel in 3 days.
The distance traveled by Jim in one day to the bus and from school = 2(8.12) = 16.24 km
Distance traveled in 3 days = 3(16.24)
= 48.72 km
Hence, he will cover 48.72 km in 3 days.
please help explain how u got the answer marking brainlist giving all my points 200 points
Answer:
v = -35
Step-by-step explanation:
-4 -31= -35 which leaves V. so v=-35
Answer:
I think that the is -27
Step-by-step explanation: 32 - (-27) = -4
Herbert is decorating the bulletin board at school.The bulletin board is 7 ft by 11 ft.He decides to add a black border.What is the length of the border needed?
Answer:
36 feets
Step-by-step explanation:
The dimension of bulletin board = 7 feets by 11 feets
Bulletin boards are shaped as a rectangle :
To add a black border ; two of the sides will be of length 7 feets (length) while the other 2 will be of 11 feets (width) each or vice-versa
However, total Lenght of black border :
(11 * 2) + (7 * 2)
22 + 14
= 36 feets
Write the coefficients of:
xin 3xy + 8y - Oy
(11) 5 in 61-57'n + 8p
(ii) z'in 3xy +42-64
(iv) yz in 3x - 5yx
Step-by-step explanation:
Here x coefficients are 3y + 8y - OyHere 5 coefficients are 61-7'n + 8pHere z coefficients are 0Here yz coefficients are 0A bricklayer lays 1200 bricks in an average 8-hour day. How many bricks does he lay in a 40-hour week?
The number of bricks that the bricklayer can lay in a 40-hour week is 6000 bricks.
Let the number of bricks that the bricklayer can lay in a 40-hour week be represented by x.
Based on the information given in the question, the equation to solve the question will be:
8/40 = 1200/x
Cross multiply
8 × x = 1200 × 40
x = (1200 × 40) / 8
x = 1200 × 5
x = 6000
Therefore, the number of bricks that the bricklayer can lay in a 40-hour week is 6000 bricks.
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what is not a composite number out of 2 4 6 8
Answer:
2
Step-by-step explanation:
2 has the factors 2 and 1 while the others have:
4- 4, 2,1
6- 6, 3, 2, 1
8- 8, 4, 2, 1
Plz help due tomorrow
Answer:
A) x is greater than or equal to 4.50
Step-by-step explanation:
the price has to be at least 4.50 so it could be 4.50 or it could be greater than 4.50. Hope this helped.
8 students in Mr. Webb's class got an A on the exit ticket. There are 25
students in his class total. What percent of students got an A?*
Answer:
32%
Step-by-step explanation:
To find the percentage, we have to multiply 25 and 8 by 4.
25x4 = 100
8x4 = 32
So, 8 out of 25 is 32%.
Three rabbits can eat five carrots in four minutes. How long does it take four rabbits to eat 15 carrots?
Answer:
12
Step-by-step explanation:
15/5=3
3x4=12
Round the 136.501 to the whole number.
Answer:
137
Step-by-step explanation:
137 because when it's .5 or higher you round up
Cash loan $100 20.0% 3 years $6 (100 x .2 x .3) $106 ($100 + $6) Big-screen TV $500 9.0% 4 years $ (500x $ College loan $13,000 4.5% 15 years $ $
Answer:
Cash loan = $160
Big screen TV = $680
College loan = $$21775
Step-by-step explanation:
Cash loan :
Loan amount or principal = $100
Interest rate, r = 20.0%
Time, t = 3 years
Using the relation :
Simple interest = Principal * rate * time
$100 * 0.2 * 3 = $60
Final amount = loan amount + interest on loan
Final amount = $100 + $60 = $160
Big Screen TV:
Principal = $500
Rate, r = 9.0%
Time, t = 4 years
Interest amount = (500 * 0.09 * 4)
Interest amount = $180
Final amount = principal amount + interest on loan
Final amount = $500 + $180 = $680
College loan
Principal, = $13,000
Rate, r = 4.5%
Time, t = 15 years
Interest amount = (13000 * 0.045 * 15)
Interest amount = $8775
Final amount = principal amount + interest on loan
Final amount = $13000 + $8775 = $21775